928. Cribbage

This problem is based on (but not identical to) the scoring for the card game Cribbage

Consider a normal pack of $52$$52$ cards. A Hand is a selection of one or more of these cards.

For each Hand the Hand score is the sum of the values of the cards in the Hand where the value of Aces is $1$$1$ and the value of court cards (Jack, Queen, King) is $10$$10$.

The Cribbage score is obtained for a Hand by adding together the scores for:

• Pairs. A pair is two cards of the same rank. Every pair is worth $2$$2$ points.

• Runs. A run is a set of at least $3$$3$ cards whose ranks are consecutive, e.g. 9, 10, Jack. Note that Ace is never high, so Queen, King, Ace is not a valid run. The number of points for each run is the size of the run. All locally maximum runs are counted. For example, 2, 3, 4, 5, 7, 8, 9 the two runs of 2, 3, 4, 5 and 7, 8, 9 are counted but not 2, 3, 4 or 3, 4, 5.

• Fifteens. A fifteen is a combination of cards that has value adding to $15$$15$. Every fifteen is worth $2$$2$ points. For this purpose the value of the cards is the same as in the Hand Score.

For example, $(5\mathrm{♠},5\mathrm{♣},5\mathrm{♢},K\mathrm{♡})$$(5 \spadesuit, 5 \clubsuit, 5 \diamondsuit, K \heartsuit)$ has a Cribbage score of $14$$14$ as there are four ways that fifteen can be made and also three pairs can be made.

The example $(A\mathrm{♢},A\mathrm{♡},2\mathrm{♣},3\mathrm{♡},4\mathrm{♣},5\mathrm{♠})$$( A \diamondsuit, A \heartsuit, 2 \clubsuit, 3 \heartsuit, 4 \clubsuit, 5 \spadesuit)$ has a Cribbage score of $16$$16$: two runs of five worth $10$$10$ points, two ways of getting fifteen worth $4$$4$ points and one pair worth $2$$2$ points. In this example the Hand score is equal to the Cribbage score.

Find the number of Hands in a normal pack of cards where the Hand score is equal to the Cribbage score.

928. 克里比奇牌戏

本题基于 克里比奇牌戏 的计分方式（但不完全相同）。

考虑一副 $52$$52$ 张牌的标准扑克。一手牌指的是该扑克中的一张或者多张牌。

对于每一手牌，其 手牌得分 的计算方式为：把该手牌中每张牌的数值相加得到的值。其中 A 的数值计为 $1$$1$、花牌（J、Q、K）的数值计为 $10$$10$。

对于每一手牌，其 克里比奇得分 是以下三项得分的总和：

• 对子：点数相同的两张牌是称为对子。每个对子计 $2$$2$ 分。

• 顺子：点数连续的 $\ge 3$$\geq 3$ 张牌称为顺子，例如：9、10、J 就是一个顺子。注意此处 A 的优先级最低，也就是说 Q、K、A 不算做顺子。每个顺子的得分是该顺子中的牌数。注意，在计分时，我们只统计极长的顺子。例如，对于 2、3、4、5、7、8、9 这手牌，我们在计分时只考虑 2、3、4、5 和 7、8、9 这两个顺子，而不考虑 2、3、4 和 3、4、5 这两个顺子。

• 十五点：若干数值之和为 $15$$15$ 的牌称为十五点。每个十五点计 $2$$2$ 分。这里数值的定义和计算手牌得分时使用的定义相同。

例如，$(5\mathrm{♠},5\mathrm{♣},5\mathrm{♢},K\mathrm{♡})$$(5 \spadesuit, 5 \clubsuit, 5 \diamondsuit, K \heartsuit)$ 的克里比奇得分是 $14$$14$，因为这手牌有 4 个十五点和 3 个对子。而 $(A\mathrm{♢},A\mathrm{♡},2\mathrm{♣},3\mathrm{♡},4\mathrm{♣},5\mathrm{♠})$$( A \diamondsuit, A \heartsuit, 2 \clubsuit, 3 \heartsuit, 4 \clubsuit, 5 \spadesuit)$ 的克里比奇得分是 $16$$16$：两个长度为 5 的顺子一共 $10$$10$ 分，两个十五点计 $4$$4$ 分，一个对子计 $2$$2$ 分。可以发现，这手牌的手牌得分恰等于克里比奇得分。

在一副 $52$$52$ 张牌的标准扑克中，有多少手牌的手牌得分恰等于克里比奇得分？

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